My only reservation is that a unit of measurement of temperature is entirely
abstract, while being in the red is not. When we *owe* money, our balance is
negative, even if the money we owe is positive...

I remember a bit about the unary minus sign and your contention that it has no
physical representation. Of course, the same could be said for the concept of
number. Counting is actually a fairly sophisticated exercise. It requires
knowledge of a number system, understanding of one-to-one correspondence, and
confidence that the quantity does not change if items are counted in a different
order. Even tallying, which does not require a number system, still requires
fairly powerful abstract reasoning. For example, if I say, "I have fourteen
chickens," the number fourteen is an abstraction. It cannot be derived from the
chickens; it has to be brought to the chickens. So fourteen (or any number, for
that matter) also has no physical representation. It could be argued that
fourteen is what all groups of fourteen items have in common, but that reasoning
is circular and neglects other properties that groups of physical items have in
common (e. g., occupies space, color, mass, duration).

Nevertheless, numbers certainly have more common applications in the lives of
students than unary minus signs do. When I introduce negative numbers, I give
examples likely to be common to everyone: below zero temperatures (especially
here in Minnesota), overdrawn bank account, and below sea level elevations. It
is especially easy to understand operations on temperatures (e. g., Most
students know that warming up 10° from −17° is −7°).

Chip Burkitt

On 8/17/2010 5:42 PM, Michael Gyori wrote:
Hello Chip,

>

>I wonder whether Ladnor might be willing to weigh in further at this poin with

>respect to the nature of equality. I don't know what a "real" mathemetician

>is, but if there is such a person, I would consider him to be one... :)

>

>As for equality, the reason I like the metaphor of a scale in discussing

>equality is that it connects numbers with physical physical phenomenon and

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